The propagation of hydromagnetic planetary waves in a rotating thin shell of fluid in regions of magnetic and velocity shears is studied using the β-plane approximation. In slowly varying shear the use of the WKBJ approximation makes it possible to construct the various types of ray trajectory that can occur and consequently the conditions that give rise to critical-latitude phenomena and trapping are deduced.

The opposite extreme to the WKBJ limit, namely reflexion and refraction of waves by a current-vortex sheet, is also analysed. In this case the conditions that lead to wave amplification (or over-reflexion) are investigated. Qualitatively, it is found that reflected Alfvén modes are amplified if the jump in the flow speed across the sheet lies between two speeds which are respectively greater and less than the sum of the Alfvén speeds on either side of the sheet. Also, Rossby waves incident upon a sufficiently strong easterly flow can suffer over-reflexion.

The general case of reflexion and refraction at a finite double (magnetic and velocity) shear layer is discussed. In analogy with the invariance of ‘wave action’ of gravity waves in a shear flow we construct a quantity [Ascr ] which is invariant except at critical latitudes, where it is discontinuous. By using the asymptotic solutions near these critical latitudes and by adopting the proper matching procedure for the solutions on either side of these latitudes it is possible to relate the two constant values of [Ascr ] on either side of each critical latitude. These general results are then applied to various profiles of shear flow and magnetic field so as to elucidate the manner in which an incident wave is reflected from and transmitted through a double layer.

An integral technique suggested for the analysis of turbulent jets by Corrsin & Uberoi (1950) and Morton, Taylor & Turner (1956) is re-examined in an attempt to improve the description of the entrainment. It is determined that the hypothesis of Priestley & Ball (1955), that the entrainment coefficient is a linear function of the jet Richardson number, is reasonable, and that two empirically determined plume parameters are sufficient to describe the transition of buoyant jets to plumes. The results of a series of experiments in which both time-averaged velocity and time-averaged temperature profiles were recorded in a substantial number of plane turbulent buoyant jets of varying initial Richardson numbers are used to verify the basic ideas. In addition, measurements of the mean tracer flux in a series of buoyant jets indicate that as much as 40% of the transport in plumes is by the turbulent flux.

The turbulence structure of a plane vertical buoyant jet in the transition state from jet-like to plume-like growth is the object of this investigation. The ambient fluid is of uniform density and motionless except for the flow induced by the jet.

An analysis of the turbulence energy equation reveals that the production of turbulent energy by the buoyancy forces relative to the production by the shear stress increases as the jet Richardson number increases, and becomes constant for a plume-like buoyant jet.

A systematic set of experiments was carried out to examine the turbulence structure for a wide range of initial Richardson numbers, extending from a value appropriate to a jet-like flow (very close to zero) to that appropriate for a plume-like flow (approximately 0·6). Fast-response thermistors and a laser-Doppler velocimeter were used to measure the buoyant jet's temperature and velocity respectively. The temperature and velocity data were recorded magnetically in digital form and subsequently processed to extract both mean and fluctuating values. The turbulence intensity and the probability density distribution of the temperature and velocity fluctuations, the maximum and minimum temperature, the intermittency, and the frequency of crossing of the hot/cold and the cold/hot interface of a buoyant jet were investigated. It was determined that the intensity of temperature and velocity fluctuations increases with increasing Richardson number. An explanation is suggested for the large-scale vortices observed in a plume.

The initial-value problem for slightly viscous, two-dimensional, spatially periodic waves is examined. Matched asymptotic expansions in space for small wave amplitude a and multiple scales in time allow the boundary layers and viscous attenuation to be described. The bottom and surface boundary layers of thickness δ are equivalent to those of Longuet-Higgins except that wave attenuation is included. For progressive waves one solution for the interior motion independent of the magnitude of δ/a is an attenuating version of the conduction solution of Longuet-Higgins, but with modified structure, the O(a2) vorticity at the boundaries ultimately diffusing into the entire field. There are certain critical depths for which there is secular behaviour and these do not correspond to quasi-steady flows. Other solutions may be possible. For standing waves the interior flow depends on the magnitude of the steady-drift Reynolds number Rs∝ (a/δ)2 introduced by Stuart. When Rs [Lt ] 1, the interior is viscous with an O(a2) vorticity ultimately diffusing into the entire field. When Rs [Gt ] 1 there is a doubleboundary-layer structure on the bottom and on the surface. Within the outer layers, the O(a2) steady drift decays to the potential flow interior. A direct analogy with the flow structure on a circular cylinder oscillating along its diameter is introduced and pursued. Finally, all of the above fields are converted to Lagrangian fields so that masstransport profiles can be obtained. Comparisons are made with previous theoretical and experimental work.

This paper presents numerical results for the added-mass and damping coefficients of semi-submerged two-dimensional heaving cylinders in water of finite depth. A simple proof is given which shows that the added mass is bounded for all frequencies in water of finite depth. The limits of the added-mass and damping coefficients are studied as the frequency tends to zero and to infinity. A new formulation valid in the low-frequency limit is constructed by using a perturbation expansion in the wavenumber parameter. For the limiting cases, dual extremum principles are used, which consist of two variational principles: a minimum principle for a functional and a maximum principle for a different but related functional. These two functionals are used to obtain lower and upper bounds on the added mass in the limiting cases. However, the functionals constructed (Bai & Yeung 1974) for the general frequency range (excluding the limiting cases) have neither a minimum nor a maximum. In this case, the approximate solution cannot be proved to be bounded either below or above by the true solution. To illustrate these methods, the added-mass and damping coefficients are computed for a circular cylinder oscillating in water of several different depths. Results are also presented for rectangular cylinders with three different beamdraft ratios at several water depths.

The stability of longitudinal rolls in an inclined convection layer is investigated for various angles of inclination. Three types of instability are responsible for the transition from longitudinal rolls to three-dimensional forms of convection in different regimes of the parameter space. The role of the wavy instability is emphasized since it does not correspond to a transition in the case of a horizontal layer. The analysis emphasizes the cases of air and water as convective media. Comparison of the theoretical results with experimental data indicates that the stability analysis based on infinitesimal disturbances correctly describes the observed instabilities.

The collapse of a composite circular viscous tube owing to external pressure and surface tension is considered. It is shown that, for small surface tension, the collapse time, at which the tube closes, is very sensitive to the viscosity of the inner tube.

This paper is the first of a pair that describe two-point velocity measurements made at various radial positions in water in fully developed pipe flow. Axial velocity fluctuations were measured with hot-film anemometers at two points sufficiently close together that the turbulence structure remained essentially unchanged while passing between them. Phases of the cross-spectra of these velocities were then determined and interpreted in terms of a wave model of the turbulence structure. The model assigns an axial velocity and streamwise inclination to the lines of equal phase of each frequency component of the spectra.

In general, the lines of equal phase for each frequency component are inclined to the wall in the flow direction, the lower frequencies being more inclined than the higher frequencies, though all lines of equal phase at points in the central region of the pipe tend towards the perpendicular. For points near the wall the inclinations are very pronounced.

In the central region, phase velocities of lower frequency components are lower than those for higher frequencies. All phase velocities could be normalized with respect to position by the local mean velocity. The group velocity of small-scale (large wavenumber) disturbances in the core region appears to be approximately constant and of the order of the local mean velocity. This leads to a modified form of Taylor's hypothesis.

The variance in all the measurements increases rapidly in the region y+ < 26. This may be due to the intermittent nature of the flow near the wall (which is discussed in part 2) or to a rotation of the ‘frozen’ pattern by the mean shear field between the two sensors. The magnitude of the latter effect is estimated in this paper and is significant very near the wall. The results in the central region are not affected.

The flow of an ostensibly two-dimensional wall jet over a logarithmic spiral has been studied both experimentally and theoretically. It is established that, if the skin friction is effectively constant, the flow may be self-preserving, and this is confirmed experimentally for the two spirals studied ($x/R = \frac{2}{3}$ and x/R = 1). The rate of growth has been predicted using the integral momentum equation and the integral equation for the combined mean and turbulent energy. Important assumptions in this theory are that the turbulence structure parameter $\overline{u^{\prime}v^{\prime}}/\overline{q^{\prime 2}}$ and the normalized mean position of the superlayer are invariant with curvature, and the experiments show that this is nearly true. The growth is constant for each spiral and increases with curvature. Using the measured rate of growth, the integral energy equation gives a satisfactory prediction of the turbulent shear stress, but the two-dimensional integral momentum does not. The turbulence is very intense in these flows and the Reynolds stresses were corrected using correlations of up to fourth order. However, the corrections may still have been too small, which would account for some of the difference between the calculated and measured shear stress. The outer flow of a wall jet strongly influences the inner boundary layer and this effect is found to increase with curvature. The conventional logarithmic law of the wall ceases to apply for $x/R >\frac{2}{3}$.

Three-dimensional homogeneous isotropic turbulence at very high Reynolds number R is studied using a variant of the Markovian eddy-damped quasi-normal theory. In the case without helicity, numerical calculations indicate the development of a $k^{-\frac{5}{3}}$ inertial range in the energy spectrum and an onset of significant energy dissipation at a time t* which appears to be independent of the viscosity v as v → 0; analytical arguments having a bearing on this behaviour, described as an ‘energy catastrophe’, are also discussed. The skewness factor (for t > t*), which increases with R, tends to 0·495 when R → ∞. When helicity is present, the existence of simultaneous energy and helicity cascades is demonstrated numerically. It is also shown that the helicity cascade inhibits the energy transfer towards large wavenumbers, in agreement with preliminary low Reynolds number results of Herring and with the conclusion of Kraichnan (1973) based on analysis of the interaction between two helicity waves. This inhibition implies a delay of the onset of energy dissipation at zero viscosity. It is shown that, whatever the relative rate of helicity and energy injection, a regime is attained at large wavenumbers k where the relative helicity tends to zero (with increasing k) and helicity is carried along locally and linearly by the energy cascade like a passive scalar. In practice, the linear regime is attained when the relative helicity is less than about 10%. The Kolmogorov constants of energy and helicity in the inertial range are determined. The impossibility of pure helicity cascades of a type conjectured by Brissaud et al. (1973a) is demonstrated. Finally it is shown that, because of dissipation and non-positive-definiteness of the helicity spectrum, non-zero total helicity may appear in the decay of unforced turbulence with zero total initial helicity, if the helicity spectrum is not initially identically zero.